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October 21, 2006

The Role of Rigour

With apologies to David Corfield, this is going to be a little philosophical disgression on the role of mathematical rigour and “proving theorems” in theoretical physics.

Back in the old days, Math 55, the honours Freshman Mathematics course at Harvard, was infamous for its “True/False” exams. The typical question involved the statement of some theorem and — if the assumptions of theorem were not stated precisely correctly — then the correct answer was “False.” This was both a brutally difficult test of the students’ mathematical knowledge and a useful object lesson. A theorem is only as good as the assumptions underlying it.

This is particularly important in Physics, where we are typically not at liberty to “redefine the problem” so that the assumptions of the theorem are satisfied.

I was reminded of this lesson by two recent discussions in the blogosphere.

Rehren Duality

The first, of course, is our recent discussion of Rehren Duality, where a purported “Theorem,” establishing an isomorphism between a QFT in AdSd+1d+1 and a conformal QFT on the dd-dimensional boundary of AdS, rests on hopelessly flawed physical assumptions.

These flaws were evident after 5 minutes, flipping through the paper. I took far longer reading it carefully, trying to reassure myself that I hadn’t somehow misconstrued what Rehren was saying. And I spent even longer, trying to make my post as clear as possible. (From the ensuing discussion, I’m not sure how well I succeeded.)

I went through this effort, not out of any sudden rekindling of interest in an obscure 7 year old paper, nor because I have something against Professor Rehren (when we met in person, he seemed like a very nice guy). I did it because there continue to be people1 who go around claiming that Rehren’s “Theorem” implies that there must be something wrong with the Maldacena Conjecture. Even that would be have been pretty much ignorable (after all, there are plenty of wrong papers on the arXivs), were it not for the peculiar amplifying nature of the internet that has, apparently, given these ideas a certain currency among impressionable young students.

That there’s an abundance of bad information on the internet is not a surprise. The troubling aspect is the totemic power of the word “Theorem,” and its ability (in the minds of some) to trump sound physical arguments and abundant calculational evidence. And it’s applied in a particularly perverse way, here, because this “Theorem” is contrasted with alleged lack of a rigourous definition of “String Theory in AdS.”

At present, AdS/CFT provides a definition of nonperturbative String Theory in asymptotically anti-de Sitter spacetimes2. The observables of the theory are defined to be the correlation functions of a certain QFT on the boundary. The Conjecture (supported by all those aforementioned calculations) is that, in appropriate limits, the resulting bulk theory reduces to semiclassical supergravity (or, in other circumstances, to a weakly-coupled string theory), and that other features that a theory of quantum gravity ought to possess do, in fact, emerge from this (somewhat unintuitive) definition.

Perhaps, someday, a better formulation of nonperturbative string theory will emerge, and we will then be able to demote the role of AdS/CFT from defining what we mean by nonperturbative string theory in AdS, to something derivable from this more fundamental formulation. But that alternative formulation will be preferred, not on the basis of its greater mathematical rigour, but on the basis of its greater explanatory power.

Helling-Policastro

Another example is discussed in a recent post by Robert Helling. Robert discusses the “polymer representation” of the spatial diffeomorphism constraints in Quantum Gravity. This odd-looking, non-separable, Hilbert space is what appears in Loop Quantum Gravity. And there is much highly technical analysis attached to it, along with many “rigourous results.” There’s even a theorem to the effect that the polymer state is unique.

It has been argued that, because of this uniqueness theorem, any background-independent quantization of gravity must proceed via this polymer representation on its “kinematical Hilbert space”.

Of course, the problem is that the assumptions of this theorem are vastly too strong. It requires that there be a state, |0⟩|0\rangle, invariant under the entire group of spatial diffeomorphisms, whereas we usually assume only that the generators of spatial diffeomorphisms have vanishing matrix elements between physical states.

Moreover, we already know that there are counterexamples to this “Theorem”. 2+1 dimensionalgravity can be quantized without invoking the polymer representation. And AdS/CFT sidesteps the whole procedure, by constructing directly the full, background-independent, quantum theory, with its set of observables, without recourse to the intermediate step of a “kinematical Hilbert Space”.

Moreover, as Robert shows in his paper, familiar systems quantized using this polymer representation seem to yield incorrect physical results.

Again, the problem is not that the Theorem is wrong, in some technical sense, but rather that its assumptions don’t (necessarily) hold in the physical systems of interest.

And Yet …

All of this is not to say that there is no place for mathematical rigour in Physics, or even in its more “speculative” areas, like String Theory and Quantum Gravity. By struggling to find a mathematically precise formulation, one often discovers facets of the subject at hand that were not apparent in a more casual treatment. And, when you succeed, rigourous results (“Theorems”) may flow from that effort.

But, particularly in more speculative subject, like Quantum Gravity, it’s simply a mistake to think that greater rigour can substitute for physical input. The idea that somehow, by formulating things very precisely and proving rigourous theorems, correct physics will eventually emerge simply misconstrues the role of rigour in Physics.

1 There’s much other drolly comment-worthy material in Schroer’s manifesto. But I’d like to ask readers to please refrain from indulging that temptation. The comment section of this post is likely to be wooly enough, as is, without degenerating completely into a fruitless discussion of Schroer’s missive.

2 Of course, this definition didn’t emerge out of thin air. It came from looking at the near-horizon geometry of a stack of D-branes in flat space. And the identification of which boundary field theory should correspond to which asymptotically-AdS string compactification comes from precisely such considerations. Still, we have no perturbative (let alone nonperturbative) formulation of string theory in most of these backgrounds. So, despite its origin in other, related, backgrounds that we do understand, AdS/CFT is, here, giving us a definition of string theory in AdS.

Posted by distler at October 21, 2006 11:07 AM

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Re: The Role of Rigour

All of this is not to say that there is no place for mathematical rigour in Physics, or even in its more “speculative” areas, like String Theory and Quantum Gravity.

I would say it differently. In my view, if you do not have experiments, then rigor is all that you have. Without experiments or rigor, research reduces to an obscure kind of art, kind-of like Horgan’s pessimistic (and frankly offensive) characterization of “the end of science”. At the moment, the connection between string theory and experiment is inadequate; its real strength comes from partial rigor.

However (this one is my however), rigor does not have to a connected structure built out from axioms. That is what it should be eventually, but that is not what it has to be as a work in progress. To build a bridge, you do not have to start at one end and add planks to get to the other side. You can plant caissons in the middle, you can connect one tower to another, and you can even let a few things fall down.

If your philosophy is to only build from one end and never let anything significant fall down, you end up building a lot of bridges to nowhere. You will also build a lot of half-bridges that cannot be extended because they would fall down. Sometimes it seems to me that mathematicians make these mistakes too often out of loyalty to rigor. (But then, sometimes it seems defensible.)

You (Jacques) have argued that Rehren duality is an example of the former (at least as it has been misread by some people), while loop quantum gravity is an example of the latter. I am not expert on these points, but it seems likely that you are right.

Meanwhile string theory looks like a very large and important bridge between physical reality and mathematical axioms, even if it is grossly incomplete, inadequately anchored at both ends, and only connected in patches.

Re: The Role of Rigour

You (Jacques) have argued that Rehren duality is an example of the former (at least as it has been misread by some people), while loop quantum gravity is an example of the latter. I am not expert on these points, but it seems likely that you are right.

Well, I’m not quite sure how to fit what I’m saying into this Civil Engineering metaphor. But I should clarify that I am talking about three somewhat distinct issues.

There’s the issue of the theorem itself, and whether the assumptions that went into it are physically-justified.

There’s the issue of a certain style of doing Physics which values proving theorems over other ways of arriving at physical knowledge.

There’s the rhetorical use to which the (alleged) theorem is put, in arguing for or against some particular approach. In particular, there’s the unreflective notion that a theorem trumps any other sort of evidence.

Of course, if you are a strong believer in (2), you are more likely to engage in arguments of the form (3), since they are the sort of arguments you, yourself, are likely to find most convincing.

Having said all that, there is clearly a qualitative difference between Rehren’s Theorem and the LOST Theorem. We can be quite certain that the former is completely physically irrelevant. In the case of the latter, the question is still open (but the work of people like Helling and Policastro makes it seem rather unlikely).

Re: The Role of Rigour

There’s the issue of a certain style of doing Physics which values proving theorems over other ways of arriving at physical knowledge.

Sure, I don’t endorse that style. But what I would say is that experiment is hardly available as one of the “other ways” in the case of string theory, or any other hypothetical theory of quantum gravity. All that you have left in this case is rigor of a kind. My point is that it doesn’t have to be theorems; rigor can instead arrive in patches. Physicists call these patches of rigor “consistency checks”.

What this discussion is about is some consistency checks which may be very strong, versus some theorems or conjectures which may be ineffectual. In my view, the extent to which string theory is “physically relevant”, in the sense of relevant to existing experiments, is sometimes overstated. Of course it’s motivated by experiments — and by see-your-hand-in-front-of-you “experiments”, things like the fact that spacetime is manifested as a Minkowskian 4-manifold. But that’s not really the same as physics the way that Enrico Fermi did physics. String theory is mostly a kind of mathematical physics, simply with a different style of rigor from much of mathematics.

Re: The Role of Rigour

Dear Jacques,

‘By making the key discoveries of their era, uneducated technicians like Michael Faraday and James Watt threatened the scholastic myth, that all progress, including scientific progress, needs must use the rigour and discipline controlled by academics in places … The ultimate in scientific rigour (rigor mortis?) was held to be mathematics. … lacking mathematics, Faraday could not and did not really effect his discovery of electromagnetic induction. Rather, he stumbled into it, but it could only be properly exploited decades later, after Professor Maxwell had placed a mathematical structure upon Faradays fumbling, unscholarly ideas. … The deeper message in Maxwell’s Equations is that, do what they will, the local yokels will not replace mathematical academia as the fount of knowledge and progress.’

However, Catt fails on substance. See for example http://www.ivorcatt.org/icrwiworld80mar1.htm where he takes merely two out of twenty long-hand differential Maxwell equations, shows that they contain two vital constants, then claims that he has proved that “Maxwell’s equations lack content” and finally sneers that Minkowski spacetime is wrong (it might be, but not for Catt’s reasons).

This may be an ironic example of the need for mathematical discipline, and the disaster which occurs when someone lacking it tries to debunk a subject. On the other hand, there are genuine problems with Maxwell, but they are more subtle and the fault is not centred on the mathematical structure, but upon the problem that the Yang-Mills field is a modification to the Maxwell field for electromagnetism. It is a shame that the crucial mathematics for key ideas in physics is so technical.

It certainly introduces the temptation for some people to misuse it, by obfuscating instead of explaining physics clearly. Even outside of string theory, large areas of physics seem to less than rigorously understood.

Re: The Role of Rigour

“Even outside of string theory, large areas of physics seem to be less than rigorously understood.”

Such as? I would say the fairly recent break with mathematicians that physicists had was sometime around the advent of field theory, virtually the entire rest of physics is about as mathematically pleasing as you can get, with only a few isolated problems that remain ‘heuristic’. Even items that were somewhat ill defined to begin with (say the Dirac Delta function), were later found to be perfectly sensible when mathematicians caught up (distribution theory).

In modern times it seems the two fields have converged again with important stringy inspired math, along with TQFTs, graded ‘susy’ algebras and so forth. So while field theory isn’t quite set in mathematical stone, its a lot closer than it used to be and many people hope it will one day be made rigorous (and you can win 1 million dollars while you’re at it)

Re: The Role of Rigour

Hi Haelfix,

I’m thinking about the supposed conflict between quantum mechanics and classical physics, see comments at http://asymptotia.com/2006/10/16/not-in-tower-records/ for instance.

Bohr simply wasn’t aware that Poincare chaos arises even in classical systems with 2+ bodies, so he foolishly sought to invent metaphysical thought structures (complementarity and correspondence principles) to isolate classical from quantum physics. This means that chaotic motions on atomic scales can result from electrons influencing one another, and from the randomly produced pairs of charges in the loops within 10^{-15} m from an electron (where the electric field is over about 10^20 v/m) causing deflections. The failure of determinism (ie closed orbits, etc) is present in classical, Newtonian physics. It can’t even deal with a collision of 3 billiard balls:

‘… the ‘inexorable laws of physics’ … were never really there … Newton could not predict the behaviour of three balls … In retrospect we can see that the determinism of pre-quantum physics kept itself from ideological bankruptcy only by keeping the three balls of the pawnbroker apart.’

Re: Quantum Chaos

Dear Jacques,

Feynman said that the Schroedinger equation wasn’t rigorously derived in 1963, and it still isn’t. Chaos theory (like string) is an example of an area of mathematics that has produced far more hype in the media than solid contributions to particle physics. (Even the “butterfly effect” on meteorology is basically a myth because small scale disruptions dissipate entirely within a short distance - the atmosphere is usually stable enough that you need wide expanses of warm sea or ground to start up storms up by convection currents.)

I’m glad people have been investigating string theory in depth, because it will prevent the same mistakes being made again based on the unproved belief that universe is mathematically elegant.

Re: Quantum Chaos

Derived from what? Hamiltonian time-evolution (the Schrœdinger equation) is usually taken to be an axiom of (nonrelativistic) quantum mechanics.

I was just thinking of saying that.

Perhaps this is only a sleepless night talking, but I’m unclear on why a “rigorous derivation” of an equation which has limited applicability would be so important. Presumably, whatever more “fundamental” postulate from which the Schrœdinger Equation could be derived would also describe only a non-relativistic world.

Re: Quantum Chaos

Dear Jacques,

Derived from physical facts, i.e., facts which are already established.

The Hamiltonian time evolution should be derived from electromagnetism: Maxwell’s displacement current basically describes energy flow (not real charge flow) due to a time varying electric field. Clearly it is wrong because the vacuum doesn’t polarize below the IR cutoff which corresponds to 10^20 volts/metre, and you don’t need that electric field strength to make capacitors, radios, etc. work.

So you could derive the Schroedinger from a corrected Maxwell ‘displacement current’ equation. This is just an example of what I mean by deriving the Schroedinger equation. Alternatively, a computer Monte Carlo simulation of electrons in orbit around a nucleus, being deflected by pair production in the Dirac sea, would provide a check on the mechanism behind the Schroedinger equation, so there is a second way to make progress.

Re: Quantum Chaos

The Hamiltonian time evolution should be derived from electromagnetism: Maxwell’s displacement current basically describes…

So you could derive the Schroedinger from a corrected Maxwell ‘displacement current’ equation. This is just an example of what I mean by deriving the Schroedinger equation. Alternatively, a computer Monte Carlo simulation of electrons in orbit around a nucleus, being deflected by pair production in the Dirac sea, …

No.

You cannot derive quantum mechanics from classical physics. And, if you want to study the quantum theory of electrons interacting with electromagnetism, you need to be doing QED, not nonrelativistic quantum mechanics.

But this is, indeed, careening off-topic. So maybe we should end it here.

Re: Quantum Chaos

Alternatively, a computer Monte Carlo simulation of electrons in orbit around a nucleus, being deflected by pair production in the Dirac sea, would provide a check on the mechanism behind the Schroedinger equation, so there is a second way to make progress.

I imagine this would produce a lot of neutrons before it produced any insight into the Schroedinger equation.

Re: Quantum Chaos

Obsessive Maths Freak, please see nige.wordpress.com - within 10^-15 metre from an electron pair production occurs because the field is above 10^20 v/m. This causes deflections of the motion of the electron because the pairs appear randomly. It doesn’t create neutrons. For pair production from QFT see http://arxiv.org/abs/hep-th/0510040 page 85 for instance.

Quackery

Re: The Role of Rigour

Four quick remarks:

1. In any applied application of mathematics modeling and interpretation are as important as rigourous theorems. Misinterpretation and overinterpretation of mathematical theorems (and even of mathematical notions) are common mistakes in application of mathematics, and in using mathematical formulations. (The issue of mis/over interpretation is important also in pure mathematics.)

2. There are areas of applications ofmathematics were analytic and regourous proofs are well beyond reach and heuristic methods, numerics and simulations are crucial instead, or in addition to, regerous theorems.

3. But there is also the flip side. A lot of modern physics including extremely successful theories are not supported by regerious mathematics. Along with the possibility that mathematics is not sufficiently developed, there is also the possibility that the non-regerious methods hide some extra physics assumptions (and even the more radical possibility that the modeling itself should be corrected). Therefore, a truly successful regerious mathematics (e.g.) for QED/QCD is something which may effect the nature of more speculative theories. The imput coming from mathematicians who study this notorious question (in the slow piece-wise-constant paste and peace-wise nature of mathematics) may be as exciting to physics as empirical evidence from LHC.

4. A strange feature of the non regerious mathematical nature of modern physics is that mathematical analysis plays there a smaller role compared to algebra and geometry.

Re: The Role of Rigour

Re: The Role of Rigour

Thanks, Jacques. I really appreciate the effort that you go to with your posts. The time wasting that results from hype is always frustrating, but I guess we just keep learning from it. I’m quite interested in AdS/CFT now.

Re: The Role of Rigour

Re: The Role of Rigour

I’d like to second Kea’s vote of confidence and add my own expression of gratitude. Good job all around — and thanks also for Planet Musings.

My only regret is that we won’t see Musings: The Book prominently displayed on the science-tabloid shelf alongside The Not-Even Elegant Trouble With the Universe (or whatever else is being hawked there now).

Re: The Role of Rigour

A couple of very educational posts.

Perhaps some day you can tell us about ADS/CFT and the role it plays in the resolution of black hole information paradox, which is not as widely understood or appreciated(for instance, how it resolves it while preserving fundamental aspects of quantum mechanics like unitarity)….

Another area of ignorance among non-experts(including myself) is how far black hole entropy results in string theory extend beyond the extremal case…

Re: The Role of Rigour

What Gina said about misapplying formalism in physical applications makes so much sense. I really have been going over these LQG papers and can see a bit of this behavior. For example, in the shadow states paper, and Theimann’s paper on the string quantization, Fell’s theorem was invoked. I read the paper and thought that Fell’s theorem is neat and everything but the application of it to LQG is questionable.

This is simply because (from what I understand) Fell’s paper doesn’t give a procedure for figuring out what the approximated/deformed H(epsilon) should be, so that is chosen arbitrarily and with seemingly prior knowledge. When the p and p^2 operators aren’t defined in the C*Algebra, yet we need those operators all of a sudden we invoke this approximate Hamiltonian. How in tarnation do we get it?

In this instance, I DO think that the mathematics has been used out of context…But I am still learning about this stuff. Maybe I am missing something?

I am quite certain that these issues have been discussed ad nauseum in other forums, so I guess I was venting a bit of frustration. Have a nice week:)

Re: The Role of Rigour

As a mathematician, I can tell you that the primary effect of mathematical rigor on physics is to stifle the subject. Let me explain.

Physicists rarely move very deeply into the mathematical rigor of any specific problem, highly theoretical physics aside. Mathematicians unfortunately, tend to take something, generalise it, compress it down to the most compact and terse form possible(which usually involves a modern esoteric branch of maths). Once they’ve done this, they represent this compacted form as a replacement for the original physical insight.

Needless to say, this axiomatic presentation of physics, and indeed mathematics, is incorrect from the start. It assumes fundamental properties and derives observed behaviours from them, rather than the way science is supposed to work, observe behaviour and derive basic properties. Honestly, I view such axiomatic presentations as being akin to the old Greek method of doing science; Devoid of experiments, properties are assummed to be “concordant” with “logic”, and any observed phenomena that do not agree with predictions are ignored.

Nowhere is this more evident than in electromagnetism and Maxwell’s equations. To this day, many, many author still use only partial time derivatives in Maxwell’s equations when they should be using total or “convective” time derivatives. The total derivatives come out naturally when one begins from the results of the physical experiments, Farday’s and Ampere’s, but many people never started this way. They learned the vector form of Maxwell’s equations as an axiom, and obtain incorrect results when dealing with moving loops.

If you’re working with physical problems, mathematical axioms won’t cut it. You need insight into the problem, otherwise the symbols you’re writing down may as well be nonsense, because by the time you finish, you won’t know what any of them “really” mean.

Re: The Role of Rigour

Charle Seife in ‘ZERO: the Biography of a Dangerous Idea’ suggests that Newton [fluxions] and Leibnitz [calculus] were in essence dividing by zero which is more consistent with insight than rigour.

Apparently in the 12th century, Indian mathematician Bhaskara used a differential calulus. Not until limit theory was developed by 19th century methematicians [Karl Theodor Wilhelm Weierstrauss?] was this insight converted to rigour.
http://en.wikipedia.org/wiki/Limit_of_a_function

Re: The Role of Rigour

I’ve belatedly stumbled upon this discussion, and am a little bemused as to why I’m being apologised to in the first line of the post. I take the topic to be philosophically interesting and care is being taken to make important distinctions. And I certainly don’t hold that only philosophers can say philosophically interesting things. So well done.

Something I’ve wondered about, and would like to hear people’s opinions on, is whether one can flip your point around. I agree that there’s a level of physical intuition that may not fit well with the rigorised mathematical treatment of the time, with the consequences outlined. But don’t we also see times when an unrigorous mathematical intuition, beyond any formalism, supplements physical intuition. A case study a student once wrote for me looked at the four way analogy Kelvin (Thomson) located in his studies of electricity, magnetism, heat, and fluids. Annoyingly I can’t quite remember which concerned flow and which statics. Anyway, the point was that an unrigorous mathematical intuition crucially aided his physical theorising.

Of course, drawing a line between physical and mathematical intuition’s going to be hard. But if one says there’s an excess on the physical side, one ought to say the same on the mathematical side.

The Role of Intuition

Your comment could be titled “The Role of Intuition,” which would make the subject of a very interesting post, in its own right. One of the annoying features of much of the Mathematics literature is that the intuition behind the results presented is carefully airbrushed out of the presentation.

Re: The Role of Intuition

So we have mathematical and physical intuition and mathematical rigour. Can we complete the square and think of ‘physical rigour’, a formalism which might strike mathematicians as not quite right, but which allows for calculations in physics. The best paper for potential examples is Pierre Cartier’s wonderful Mathemagics (A Tribute to L. Euler and R. Feynman).

Where we hear so much about mathematical rigour and physical intuition, the other two corners of the square are very important, and the ways in which each pair of corners can interact are multifarious, some productive and some not.

Physical Rigour

You raise an interesting question. Most often, “a physicist’s level of rigour” or “a physics proof” are taken to be somewhat pejorative descriptions, simply inferior versions of what a mathematician would consider satisfactory.

But is there a distinctive style that might be called “physical rigour”?

There are probably many facets that one could identify; here’s one.

As you know, there’s a Clay prize for proving the existence of a mass gap in quantum Yang Mills theory. It is considered a great unsolved problem. On the other hand, the lattice gauge theorists have computed, numerically, the spectrum of low-lying excitations to a pretty high degree of accuracy (a few percent). That is, their computations tell you much more than the mere existence of a gap in the spectrum.

But, somehow, those numerical results don’t “count.” Of course, even we physicists would love to have an analytical understanding of the mass gap. But I think our attitude towards the numerical results is rather different. I think they constitute a fine example of (one kind of) “physical rigour.”